Solved on Mar 06, 2024

Simplify the expression: $\tan \left(\frac{\pi}{2}-x\right) \sec x$

STEP 1

Assumptions
1. The trigonometric identity for tangent of the difference of two angles is $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$ .
2. The specific trigonometric identities for $\tan \left(\frac{\pi}{2}\right)$ and $\tan x$ will be used.
3. The trigonometric identity for $\sec x$ is the reciprocal of $\cos x$ , which means $\sec x = \frac{1}{\cos x}$ .
4. The relationship between the trigonometric functions of complementary angles will be used, specifically that $\tan \left(\frac{\pi}{2} - x\right) = \cot x$ .

STEP 2

Apply the identity for the tangent of the difference of two angles to the expression $\tan \left(\frac{\pi}{2}-x\right)$ .
$\tan \left(\frac{\pi}{2}-x\right) = \frac{\tan \left(\frac{\pi}{2}\right) - \tan x}{1 + \tan \left(\frac{\pi}{2}\right) \tan x}$

STEP 3

Recognize that $\tan \left(\frac{\pi}{2}\right)$ is undefined because the tangent function approaches infinity as its argument approaches $\frac{\pi}{2}$ from the left. However, in the context of the given expression, we are interested in the behavior of the tangent function as it relates to the cotangent function, which is defined as $\cot x = \frac{1}{\tan x}$ .

STEP 4

Use the relationship between the tangent and cotangent functions for complementary angles.
$\tan \left(\frac{\pi}{2}-x\right) = \cot x$

STEP 5

Since $\cot x = \frac{1}{\tan x}$ , we can rewrite the expression as:
$\tan \left(\frac{\pi}{2}-x\right) = \frac{1}{\tan x}$

STEP 6

Now, we multiply the expression $\tan \left(\frac{\pi}{2}-x\right)$ by $\sec x$ .
$\tan \left(\frac{\pi}{2}-x\right) \sec x = \frac{1}{\tan x} \sec x$

STEP 7

Substitute the identity for $\sec x$ as the reciprocal of $\cos x$ .
$\frac{1}{\tan x} \sec x = \frac{1}{\tan x} \cdot \frac{1}{\cos x}$

STEP 8

Recognize that $\tan x = \frac{\sin x}{\cos x}$ and rewrite the expression in terms of sine and cosine.
$\frac{1}{\tan x} \cdot \frac{1}{\cos x} = \frac{1}{\frac{\sin x}{\cos x}} \cdot \frac{1}{\cos x}$

STEP 9

Simplify the expression by multiplying the denominators.
$\frac{1}{\frac{\sin x}{\cos x}} \cdot \frac{1}{\cos x} = \frac{\cos x}{\sin x} \cdot \frac{1}{\cos x}$

STEP 10

Cancel out the common factor of $\cos x$ in the numerator and denominator.
$\frac{\cos x}{\sin x} \cdot \frac{1}{\cos x} = \frac{1}{\sin x}$

STEP 11

Recognize that $\frac{1}{\sin x}$ is the definition of $\csc x$ .
$\frac{1}{\sin x} = \csc x$

STEP 12

Conclude the simplification of the original expression.
$\tan \left(\frac{\pi}{2}-x\right) \sec x = \csc x$
The simplified expression is $\csc x$ .

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Simplify the expression: $\tan \left(\frac{\pi}{2}-x\right) \sec x$

STEP 1

STEP 2

Apply the identity for the tangent of the difference of two angles to the expression $\tan \left(\frac{\pi}{2}-x\right)$ .
$\tan \left(\frac{\pi}{2}-x\right) = \frac{\tan \left(\frac{\pi}{2}\right) - \tan x}{1 + \tan \left(\frac{\pi}{2}\right) \tan x}$

STEP 3

STEP 4

Use the relationship between the tangent and cotangent functions for complementary angles.
$\tan \left(\frac{\pi}{2}-x\right) = \cot x$

STEP 5

Since $\cot x = \frac{1}{\tan x}$ , we can rewrite the expression as:
$\tan \left(\frac{\pi}{2}-x\right) = \frac{1}{\tan x}$

STEP 6

Now, we multiply the expression $\tan \left(\frac{\pi}{2}-x\right)$ by $\sec x$ .
$\tan \left(\frac{\pi}{2}-x\right) \sec x = \frac{1}{\tan x} \sec x$

STEP 7

Substitute the identity for $\sec x$ as the reciprocal of $\cos x$ .
$\frac{1}{\tan x} \sec x = \frac{1}{\tan x} \cdot \frac{1}{\cos x}$

STEP 8

Recognize that $\tan x = \frac{\sin x}{\cos x}$ and rewrite the expression in terms of sine and cosine.
$\frac{1}{\tan x} \cdot \frac{1}{\cos x} = \frac{1}{\frac{\sin x}{\cos x}} \cdot \frac{1}{\cos x}$

STEP 9

Simplify the expression by multiplying the denominators.
$\frac{1}{\frac{\sin x}{\cos x}} \cdot \frac{1}{\cos x} = \frac{\cos x}{\sin x} \cdot \frac{1}{\cos x}$

STEP 10

Cancel out the common factor of $\cos x$ in the numerator and denominator.
$\frac{\cos x}{\sin x} \cdot \frac{1}{\cos x} = \frac{1}{\sin x}$

STEP 11

Recognize that $\frac{1}{\sin x}$ is the definition of $\csc x$ .
$\frac{1}{\sin x} = \csc x$

STEP 12

Conclude the simplification of the original expression.
$\tan \left(\frac{\pi}{2}-x\right) \sec x = \csc x$
The simplified expression is $\csc x$ .

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Simplify the expression: tan⁡(π2−x)sec⁡x\tan \left(\frac{\pi}{2}-x\right) \sec xtan(2π​−x)secx

STEP 1

STEP 2

STEP 3

STEP 4

Use the relationship between the tangent and cotangent functions for complementary angles.tan⁡(π2−x)=cot⁡x\tan \left(\frac{\pi}{2}-x\right) = \cot xtan(2π​−x)=cotx

STEP 5

Since cot⁡x=1tan⁡x\cot x = \frac{1}{\tan x}cotx=tanx1​, we can rewrite the expression as:tan⁡(π2−x)=1tan⁡x\tan \left(\frac{\pi}{2}-x\right) = \frac{1}{\tan x}tan(2π​−x)=tanx1​

STEP 6

Now, we multiply the expression tan⁡(π2−x)\tan \left(\frac{\pi}{2}-x\right)tan(2π​−x) by sec⁡x\sec xsecx.tan⁡(π2−x)sec⁡x=1tan⁡xsec⁡x\tan \left(\frac{\pi}{2}-x\right) \sec x = \frac{1}{\tan x} \sec xtan(2π​−x)secx=tanx1​secx

STEP 7

Substitute the identity for sec⁡x\sec xsecx as the reciprocal of cos⁡x\cos xcosx.1tan⁡xsec⁡x=1tan⁡x⋅1cos⁡x\frac{1}{\tan x} \sec x = \frac{1}{\tan x} \cdot \frac{1}{\cos x}tanx1​secx=tanx1​⋅cosx1​

STEP 8

STEP 9

Simplify the expression by multiplying the denominators.1sin⁡xcos⁡x⋅1cos⁡x=cos⁡xsin⁡x⋅1cos⁡x\frac{1}{\frac{\sin x}{\cos x}} \cdot \frac{1}{\cos x} = \frac{\cos x}{\sin x} \cdot \frac{1}{\cos x}cosxsinx​1​⋅cosx1​=sinxcosx​⋅cosx1​

STEP 10

Cancel out the common factor of cos⁡x\cos xcosx in the numerator and denominator.cos⁡xsin⁡x⋅1cos⁡x=1sin⁡x\frac{\cos x}{\sin x} \cdot \frac{1}{\cos x} = \frac{1}{\sin x}sinxcosx​⋅cosx1​=sinx1​

STEP 11

Recognize that 1sin⁡x\frac{1}{\sin x}sinx1​ is the definition of csc⁡x\csc xcscx.1sin⁡x=csc⁡x\frac{1}{\sin x} = \csc xsinx1​=cscx

STEP 12

Conclude the simplification of the original expression.tan⁡(π2−x)sec⁡x=csc⁡x\tan \left(\frac{\pi}{2}-x\right) \sec x = \csc xtan(2π​−x)secx=cscxThe simplified expression is csc⁡x\csc xcscx.

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Simplify the expression: $\tan \left(\frac{\pi}{2}-x\right) \sec x$

Use the relationship between the tangent and cotangent functions for complementary angles.
$\tan \left(\frac{\pi}{2}-x\right) = \cot x$

Since $\cot x = \frac{1}{\tan x}$ , we can rewrite the expression as:
$\tan \left(\frac{\pi}{2}-x\right) = \frac{1}{\tan x}$

Now, we multiply the expression $\tan \left(\frac{\pi}{2}-x\right)$ by $\sec x$ .
$\tan \left(\frac{\pi}{2}-x\right) \sec x = \frac{1}{\tan x} \sec x$

Substitute the identity for $\sec x$ as the reciprocal of $\cos x$ .
$\frac{1}{\tan x} \sec x = \frac{1}{\tan x} \cdot \frac{1}{\cos x}$

Simplify the expression by multiplying the denominators.
$\frac{1}{\frac{\sin x}{\cos x}} \cdot \frac{1}{\cos x} = \frac{\cos x}{\sin x} \cdot \frac{1}{\cos x}$

Cancel out the common factor of $\cos x$ in the numerator and denominator.
$\frac{\cos x}{\sin x} \cdot \frac{1}{\cos x} = \frac{1}{\sin x}$

Recognize that $\frac{1}{\sin x}$ is the definition of $\csc x$ .
$\frac{1}{\sin x} = \csc x$

Conclude the simplification of the original expression.
$\tan \left(\frac{\pi}{2}-x\right) \sec x = \csc x$
The simplified expression is $\csc x$ .